Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

For what values of \(a, b\), and \(c\) are the three vectors \(\langle a, 0,1\rangle,\langle 0,2, b\rangle\), and \(\langle 1, c, 1\rangle\) mutually orthogonal.

Step-by-Step Solution

Answer

The vectors are mutually orthogonal when \(a = -1\), \(b = 0\), and \(c = 0\).

Step 1: Understanding Orthogonality

Two vectors are orthogonal if their dot product is zero. We need to check the dot products of all pairs of the given vectors: \( \langle a, 0, 1 \rangle \), \( \langle 0, 2, b \rangle \), and \( \langle 1, c, 1 \rangle \).

Step 2: Dot Product of First Pair

Calculate the dot product of \( \langle a, 0, 1 \rangle \) and \( \langle 0, 2, b \rangle \):\[ (a \cdot 0) + (0 \cdot 2) + (1 \cdot b) = b \]For orthogonality, set the dot product to zero: \[ b = 0 \]

Step 3: Dot Product of Second Pair

Calculate the dot product of \( \langle a, 0, 1 \rangle \) and \( \langle 1, c, 1 \rangle \):\[ (a \cdot 1) + (0 \cdot c) + (1 \cdot 1) = a + 1 \]Set the dot product to zero for orthogonality:\[ a + 1 = 0 \]Solve for \( a \): \[ a = -1 \]

Step 4: Dot Product of Third Pair

Calculate the dot product of \( \langle 0, 2, b \rangle \) and \( \langle 1, c, 1 \rangle \):\[ (0 \cdot 1) + (2 \cdot c) + (b \cdot 1) = 2c + b \]Using the value \( b = 0 \) from Step 2, we simplify:\[ 2c + 0 = 0 \]Solve for \( c \):\[ 2c = 0 \]\[ c = 0 \]

Step 5: Conclusion

Using the calculations from each pair, we determine the values for which all vectors are mutually orthogonal: \( a = -1 \), \( b = 0 \), and \( c = 0 \).

Key Concepts

Vectors

Vectors are fundamental objects in multivariable calculus that represent a quantity with both magnitude and direction. They are often used to model physical phenomena such as forces or velocities.
In mathematical terms, a vector in three dimensions is represented as \(\langle x, y, z \rangle\).
This involves three components:

• \(x\): component along the x-axis
• \(y\): component along the y-axis
• \(z\): component along the z-axis

Vectors can be visualized as arrows pointing from one point to another in space, with the direction of the arrow indicating the direction of the vector. Understanding vectors is crucial as they form the building blocks for analyzing more complex structures and transformations in mathematics.

Orthogonality

Orthogonality is a concept from linear algebra and multivariable calculus that describes a situation where two vectors are perpendicular to each other.
If two vectors are orthogonal, their dot product is zero. This is because the angle between them is 90 degrees, which naturally means that they do not share any component in any particular dimension.
When verifying orthogonality, we calculate the dot product of two vectors and check the result:

• If the dot product is zero, the vectors are orthogonal.
• If the dot product is not zero, they are not orthogonal.

This property is integral to numerous applications, including finding the basis for a vector space, constructing projections, and solving problems involving angles and distances in geometry.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually vectors) and returns a single number.
For two vectors \(\langle a_1, b_1, c_1 \rangle\) and \(\langle a_2, b_2, c_2 \rangle\), the dot product is calculated as:\[a_1 \times a_2 + b_1 \times b_2 + c_1 \times c_2\]
The result of this operation is a scalar, hence the name scalar product. The dot product has several interpretations:

• Geometrically, it relates to the cosine of the angle between two vectors.
• Algebraically, it computes the sum of the products of the corresponding components of two vectors.
• In physical contexts, it can represent work done when a force is applied in the direction of displacement.

Understanding how to compute and interpret the dot product is essential in solving vector-related problems, particularly in determining orthogonality.